Most games will 'run'. But it depends how nicely you want them to run.

I'd avoid any vaguely recent FPSs. Framerate drops will make them unplayable and you don't really have a powerful graphics card. Turn based strategy games will be less obvious.

If you haven't already, I'd upgrade the RAM to either 8GB (2x4GB DDR3 1333MHz SODIMM) or 16GB (2x8GB). The increase in the RAM will allocate more VRAM to the integrated graphics card. It won't make a huge difference, but it's something. I'd also swap the HDD for an SSD for quicker loading times and a considerably smoother OS experience across the board. This will maximise games performance on your laptop.

Mac OS is not made for gaming. You need to install Windows if you want to play games.

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The abundance of OS X games would say otherwise. Of course you can play games on a Mac. And if you're serious about gaming, you wouldn't be doing it on a 2011 MacBook Pro with an integrated graphics card, regardless of which operating system you're running.

The OP can run many games on their Mac. Nothing like Fallout 4 of course, but that's to be expected.

Here are some that I have played on my now quite old Macbook Pro Retina:
Wesnoth
Drakensang Online
UO: Renaissance
Age of Conquest IV
League of Legends, I downloaded but never managed to get working. Probably my too slow internet connection.

More about the Ping for LoL. The point still stands that there is a tiny variety of mostly indie games one Mac. You need to install W7-10. I mean just look at the lack of content on the App Store and the Steam Mac section. Until Apple Support DX their OS will not be a viable option.

The Intel HD 3000 is very limited for gaming, but there are older games that will work. Lots of indie games will also manage on it.

Notebookcheck has a list of benchmarks for major games that will give you some examples of what will run and at what settings. Though those numbers are for Windows, and OS X will lose a few frames.

The suggestion to game on Windows will open up a larger selection of games, and the games will perform better, but I wouldn't recommend investing that way unless there are specific games you want that will support your integrated graphics.

As usual, truth is somewhere in the middle of what is posted above, or - to say it different - everyone has a point. If you care about Blizzard games, for example, you can expect double fps just by switching to windows, no matter the gpu. Other games (engines) will not be so kind with a not-so-recent integrated gpu and will suck no matter what OS you use.

Obviously, the latest the title, the less the chances to work with HD3000. Windows, however, will offer you a (vastly) larger variety of games that will - maybe - run on low settings, while on OS X side will run slower or they will not even exist. So it's your best bet, anyway. However, with a HD3000 it might not worth the windows purchase before upgrading your machine.

With the HD 3000 graphics unfortunately you're going to be really limited. Doubt you'd be able to run CS:GO on it at anything higher than the absolute lowest quality graphics and even then it'd probably struggle.

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